Using Fubini, this last one is equal to So ie. is self-adjoint.
Now suppose is an eigenfunction of with eigenvalue with as defined in the sheet then:
. In using this last expression for it's clear that (as long as , a difficulty that'll dissapear in the next step).
Now, assuming is twice differentiable (only continuity will not be enough) we have:
and so . Now just notice that we never used so if we just have which satisfies the boundary value problem trivially.